Weighted-L-2 Polynomial Approximation In C

We study the density of polynomials in H-2(Omega, e(-phi)), the space of square integrable holomorphic functions in a bounded domain Omega in C, where phi is a subharmonic function. In particular, we prove that the density holds in Caratheodory domains for any subharmonic function phi in a neighborhood of (Omega) over bar. In non-Caratheodory domains, we prove that the density depends on the weight function, giving examples.